[rckoepke]

Revisiting this. Isn't it a bit of a red herring to enquire about the number of 2-faces that an n-simplex has? It still only has n+1 vertices. A 768-simplex may have 75.5 million faces but it will still only have 769 vertices which completely define the shape. So why would I expect a large number of the other >90% of the 10,000 samples I have to lie on the surface, rather than inside the interior volume?

To be more direct, what's the specific relevance of bringing up the number of 2-faces that an n-simplex has?

[blamestross]

So you won't be able to define a hull at all without (n+1) points. He has 10k points so that sounds like a lot relatively.

So the definition of a convex hull is, put generally, the set of points that define faces such that every point is on the face or on the "inside" of it (mean point side)

To test if a point is inside that simplex hull, you need to check every one of those faces. But that isn't the problem.

Every face is a "filter" that all the other points have to pass. Moving beyond the simplex, at higher dimensions, the number of faces that need to "accept" every other point scales faster in higher dimensions. The odds of that aren't horribly clear, you are right in other comments to call out that the structure of points in this context is by definition not independent or random, but you need enough structure to get around the fact that high dimensional hulls are basically all surface.